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In this paper we associate an abelian category to a finite directed graph andprove the categories arising from two graphs are equivalent if the incidencematrices of the graphs are shift equivalent. The abelian category is thequotient of the category of graded vector space representations of the quiverobtained by making the graded representations that are the sum of their finitedimensional submodules isomorphic to zero.  Actually, the main result in this paper is that the abelian categories areequivalent if the incidence matrices are strong shift equivalent. That resultis combined with an earlier result of the author to prove that if the incidencematrices are shift equivalent, then the associated abelian categories areequivalent.  Given William's Theorem that subshifts of finite type associated to twodirected graphs are conjugate if and only if the graphs are strong shiftequivalent, our main result can be reformulated as follows: if the subshiftsassociated to two directed graphs are conjugate, then the categories associatedto those graphs are equivalent.

Shift equivalence and a category equivalence involving graded modules over path algebras of quivers